The answer is no in general: for example, if $\phi$ is a cuspidal automorphic form in a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb{A}_F)$ and $\Phi$ is an Eisenstein series in the noncuspidal automorphic representation $\Pi = \widetilde{\pi} \boxplus \omega$ of $\mathrm{GL}_n(\mathbb{A}_F)$, where $\omega$ is a Hecke character, then were this integral to converge absolutely, it would represent $L(s,\Pi \times \pi) = L(s,\widetilde{\pi} \times \pi) L(s,\pi \otimes \omega)$. However, this has a pole at $s = 1$.
On the other hand, if $\Phi$ is cuspidal, then this integral converges absolutely even if $\phi$ is an Eisenstein series. More generally, in order to relate this integral to $L$-functions, you need to use a regularisation process, which is due to Ichino and Yamana:
https://doi.org/10.1112/S0010437X14007362
The idea is to use a truncation operator on these automorphic forms that truncates up to height $T$, and then view this as a polynomial in $T$. The constant term in this polynomial is precisely what one would hope for, namely $$\int\limits_{\mathrm{N}_n(\mathbb{A}_F) \backslash \mathrm{GL}_n(\mathbb{A}_F)} W_{\Phi}\begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix} W_{\phi}(g) \left|\det g\right|^{s - \frac{1}{2}} \, dg,$$ where $W_{\Phi} \in \mathcal{W}(\Pi,\psi)$ is the Whittaker function associated to $\Phi$ and $W_{\phi} \in \mathcal{W}(\pi,\overline{\psi})$ is the Whittaker function associated to $\phi$. If these Whittaker functions are pure tensors, then this global integral factorises as a product of local integrals, each of which represents the local component of the $\mathrm{GL}_{n + 1} \times \mathrm{GL}_n$ Rankin-Selberg $L$-function.